344 MR. WARREN ON GEOMETRICAL REPRESENTATION 



then b' =: m a' ; 



'^^ " In like manner let b' =i m a'; 



"* " y 9 



then b' — b' = m a' — m a', 



X y p q 



:. (by Art. 7-) ^ — y . c ^ — I =m.p — q.cj— \, 



:.cc— yr=.m.f — q, 

 where m is either irrational or impossible ; therefore, since x, y, ]), q are either 

 = or are whole numbers either positive or negative, the conditions of the 

 equation cannot be satisfied unless p = q, 



.-. p = q. 

 21. Let (aV" = b, and let /bV = f, where a, m, n, are any quantities 



whatever ; 



then, if b' be that value of b', which is equal to m a', /aX™" =/. 



For, since / = /'i\" 



f^nb' 



z=i mn a! 

 p 



mn 



^j,.m + n>/-l _ /£ 



/EV'' (y\^, where m and n are any possible quanti- 



ties whatever, and c is the circumference of the circle, and E the base of the 







hyperbolic logarithms. 



Forlet/E\'" + "^^=f, 



then one of the values of f' is m + n ^J — 1, 

 let f' be that value of /, ^* 



then g' = w + n ^/ — 1 ; 



let r be a positive quantity, in length = f, and let f be inclined to unity at an 

 angle = 6, d being positive and less than c the circumference of the circle, 

 then f' = r' + 6 + q c . V" 1 ? 



». 



